Periodic square wave

Theoretically, the perturbation can be an arbitrary, more or less complex, function of time. However, only a limited number of functions have been shown to be of practical importance. These are known as step, pulse, double-step, double-pulse, periodic square wave and periodic sine wave. A survey of the most common techniques is found in Table 2. 

Consider one-dimensional diffusion in an infinite medium with a periodic square wave initial condition given by... 

SWV is one of the most popular electrochemical techniques, mainly in electroanalysis, due to its great sensitivity, discrimination of background currents, and short experimental times [6, 9, 12]. It was introduced by Baker [13-15] and later developed by the Osteryoungs and coworkers by using a combination of a staircase potential modulation and a periodic square wave potential function [16-19]. In the SWV technique, the potential sequence can be described as (see Scheme 7.3 and [9]) ... 

An infinite number of harmonics must be added to obtain, for instance, a true square wave. The Fourier series for a periodic square wave of unit amplitude is (see Figure 8.6(a)) ... 

We make the observation that purely sinusoidal waves have no actual physical reality and that we are invariably dealing with anhaimonic features in practice. The synthesis of a periodic square wave into harmonic components provides a good demonstration of the principle (Fig. 26.11). 

Suppose that the process to be identified is placed under relay feedback control and oscillates with some period T. Using a sampling interval of At, the number of samples within a period is AT =. The periodic square wave u k) generated by the relay output can be completely described over this period [0, T] using a discrete Fourier expansion (Godfrey, 1993)... 

Both the outlet pressure and the feed velocity were varied in order to reproduce, as a function of and /p , values, negative pulses for TMP and positive pulses for Q, according to the already-described periodic square-wave profiles. At the interface of the channel and the membrane, it was assumed that the components of the velocity vectors were continuous, while, on the basis of the osmotic pressure model, the pressure pi, evaluated on the membrane-side , was equal to the pressure p, calculated on the channel-side , decreased by the difference of osmotic pressure. All (x,t), between the membrane surface and the permeate ... 

Electrically insulating materials can be analyzed in HF-plasma SNMS by applying a square-wave HF in the 100 kHz range to the sample (Fig. 3.34). Dielectric charge transfer at the start of a period shifts the surface potential to the amplitude Uhfm applied. Ar" ions are attracted from the plasma and sputter the surface until the end of At . The potential increase AU = 1-100 V caused by their charge is then converted to a positive absolute AU which is reduced to less than 1 V within <0.1 ps by the... 

In a steady state experiment the PIA signal Y is proportional to neq. Measuring the PIA with a lock-in amplifier means exciting the sample with a periodic time-dependent pump photon flux. The latter can be approximated by a square wave that switches between a constant flux and zero photons with a frequency /= 1/r. As shown in Refs. [32] and [33] the PIA signal, measured with a lock-in amplifier Y, shows the same functional dependence on p as ncq in Eq. (9.5). For the monomo-lecular (p-1) and bimolecular (//=2) case the influence of r depends on t, the lifetime of the observed states, as follows ... 

FIGURE 3-8 Square-wave wavefonn showing the amplitude, Esw step height, AE square-wave period, r delay time, Td and current measurement times, 1 and 2. (Reproduced with permission from reference 9.)... 

Overall, the RDE provides an efficient and reproducible mass transport and hence the analytical measurement can be made with high sensitivity and precision. Such well-defined behavior greatly simplifies the interpretation of the measurement. The convective nature of the electrode results also in very short response tunes. The detection limits can be lowered via periodic changes in the rotation speed and isolation of small mass transport-dependent currents from simultaneously flowing surface-controlled background currents. Sinusoidal or square-wave modulations of the rotation speed are particularly attractive for this task. The rotation-speed dependence of the limiting current (equation 4-5) can also be used for calculating the diffusion coefficient or the surface area. Further details on the RDE can be found in Adam s book (17). 

In such a synthesis the lengths of the pulses are variable as well as the potentials of the square wave. Recently a potential-time profile has been used to maintain the activity of an electrode during the oxidation of organic compounds (Clark et al., 1972) at a steady potential the current for the oxidation process was observed to fall, but a periodic short pulse to cathodic potentials was sufficient to prevent this decrease in electrode activity. 

Here, results are shown from experiments performed in ASTER, reported by Biebericher et al. [512. 519], A SiH4-H2 (50 50 flow ratio, total flow 60 seem) plasma was generated at an RF excitation frequency of 50 MHz. The substrate temperature was 250°C. The RF signal was ampitude modulated (AM) by a square wave. The modulation frequency has been varied in a range of 1-400 kHz. The modulation depth was always 90%. The duty cycle was fixed at 50%. The pressure amounted to 0.2 mbar, and the average power was kept at 10 W. With a duty cycle of 50%, this leads to a power of 20 W during the plasma-on period. 

Above t = 60 minutes, changes in the time-average cycling rate at an amplitude of 0.24 were too small to be significant. At the other end of the frequency spectrum, a different timer would have been required to achieve periods less than 1 minute. Periods of less than 1 minute were not investigated furthermore because at shorter periods the square wave becomes increasingly distorted by mixing in the system. 

Current-voltage profiles for chlorine evolution, obtained at various times between periods of square-wave potential cycling (1.35 to — 0.32 V versus SCE, 60 s cycle-1), all at the same 40 at. % Ru electrode, are shown in Fig. 5.4. It can be seen that the activity of these electrodes increases at the beginning of the deactivation period, as revealed by the decrease in the overpotential at approximately 5 h. This may arise, at least in part, from... 

Unlike in CV, pairs of current measurements are made on each period of the square wave. These are at time forward late in the forward pulse, named /forward, and frevere in the reverse pulse, named Iteveix- Both tformad and Reverse are much greater than the time for fully charging the electrode capacitance, so that only the Faradaic current is recorded. With calculation of /net, the difference between /forward and /reverse, SWV presents three types of peak-shaped I-E relations. Figure 65 displays the SWV of a reversible one-electron reduction process. 

Sinusoidal excitation provides only one harmonic at the modulation frequency. In contrast, pulsed light provides a large number of harmonics of the excitation repetition frequency. The harmonic content, the number of harmonics and their amplitude, is determined by the pulse width and shape.(25) For example, a train of infinitely short pulses provides an infinite number of harmonics all with equal amplitude. A square wave provides only three modulation frequencies with sufficient amplitude to be usable. Equation (9.74) gives the harmonic content of a train of rectangular pulses R(t) of D duty cycle (pulse width divided by period) and RP peak value ... 

An alternative and more recent electroanalytical tool is square-wave voltammetry (which is probably now employed more often than normal or differential pulse voltammetry). In this technique, a potential waveform (see Figure 6.26) is applied to the working electrode. Pairs of current measurements are then made (depicted on the figure as t and f2) these measurements are made for each wave period ( cycle ), which is why they are drawn as times after to (when the cycle started). The current associated with the forward part of the pulse is called /forward, while the current associated with the reverse part is /reverse- A square-wave voltammogram is then just a graph of the difference between these two... 

Figure 2.31 Effect of film thickness on the absorbance response of a PXV/PEDOT PSS films bearing (a) 20 (b) 30 (c) 40 (d) 50 and (e) 60 bilayers. Square-wave potential between 0.5 V and —0.9 V and a period of 20s was employed. Taken from [196].

Fig. 1.8 Scheme of the square-wave voltammetric excitation signal, st starting potential, Esv, pulse height, AE potential increment, t staircase period, to delay time and 7f and 4 denote the points where the forward and backward currents are sampled, respectively... 

Each square-wave half-period is divided into 25 time increments d = (50/) By introducing (1.19) and (1.20) into (1.8), the following system of recnrsive formnlae is obtained ... 

Note that the product /A yields the scan rate of the square-wave potential modulation. If the delay period is sufficiently long, the additional adsorption during the potential scan is negligible. Otherwise, the additional adsorption complicates the theoretically expected dependencies, in particular the relationships between the net peak currents and potentials on the frequency [114]. 

Vpulse, IPULSE - Pulse waveform. Can be used to create a square wave by specifying the pulse width and period. 

S0LUTI0I1 Use the pulsed voltage source. Set the rise and fall times to 1 ps so that the rise and fall times are much shorter than the pulse width and period of the square wave. Wire the circuit ... 

PUL5E WIDTH - The amount of time the voltage spends at the pulsed value. It must be true that PULSEJMDTH < PERIOD. For a square wave, PULSE WIDTH = PERIOD/2... 

We would like to create a 5 V square wave at a frequency of 1 kHz. The rise and fall times will be 1 ps. We will specify the following values for the attributes Period = lm, rise time = lu, fall time = lu, pulse width = 0.5m, initial voltage = 5, pulsed voltage = -5, delay time = 0. A few of these settings are shown in the spreadsheets below ... 

In these equations, Ej represents the electrode potential during the yth half period, 5 the fraction of the square-wave half period at which the current is measured, / is the square-wave frequency (equal to the inverse of the square-wave period), and the other symbols have their customary meaning. As long as the square-wave amplitude, Esw, is lower than 0.5RT jnF—a condition easily accomplished under the usual experimental conditions—the differential sum of the currents flowing during the anodic and cathodic half cycles can be represented by an expression such as [184]... 

Fig. 3.14 Square-wave voltammograms of PlGEs modified with sample Cl-12 from abronze mon-tefortino helmet from the Gabriel river valley (Kehn and Ikalesken period) in the Valencian region of Requena, dating back to the Second Iron Age. Electrolyte 0.50 M potassium phosphate buffer, pH 7.0. Potential scan initiated at +650 mV in the negative direction. Potential step increment 4 mV square wave amplitude 25 mV frequency 5 Hz...

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